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Tag 0AZT

Chapter 42: Intersection Theory > Section 42.14: Intersection multiplicities using Tor formula

Lemma 42.14.2. Let $X$ be a nonsingular variety. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. The $\mathcal{O}_X$-module $\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent, has stalk at $x$ equal to $\text{Tor}_p^{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)$, is supported on $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and is nonzero only for $p \in \{0, \ldots, \dim(X)\}$.

Proof. The result on stalks was discussed above and it implies the support condition. The $\text{Tor}$'s are coherent by Lemma 42.14.1. The vanishing of negative $\text{Tor}$'s is immediate from the construction. The vanishing of $\text{Tor}_p$ for $p > \dim(X)$ can be seen as follows: he local rings $\mathcal{O}_{X, x}$ are regular (as $X$ is nonsingular) of dimension $\leq \dim(X)$ (Algebra, Lemma 10.115.1), hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim(X)$ (Algebra, Lemma 10.109.8) which implies that $\text{Tor}$-groups of modules vanish beyond the dimension (More on Algebra, Lemma 15.61.19). $\square$

    The code snippet corresponding to this tag is a part of the file intersection.tex and is located in lines 722–733 (see updates for more information).

    \begin{lemma}
    \label{lemma-compute-tor-nonsingular}
    Let $X$ be a nonsingular variety.
    Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules.
    The $\mathcal{O}_X$-module
    $\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$
    is coherent, has stalk at $x$ equal to
    $\text{Tor}_p^{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)$,
    is supported on
    $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and
    is nonzero only for $p \in \{0, \ldots, \dim(X)\}$.
    \end{lemma}
    
    \begin{proof}
    The result on stalks was discussed above and it implies the support
    condition. The $\text{Tor}$'s are coherent by
    Lemma \ref{lemma-tensor-coherent}. The vanishing of negative
    $\text{Tor}$'s is immediate from the construction. The
    vanishing of $\text{Tor}_p$ for $p > \dim(X)$ can be seen as follows:
    he local rings $\mathcal{O}_{X, x}$ are regular
    (as $X$ is nonsingular) of dimension $\leq \dim(X)$
    (Algebra, Lemma \ref{algebra-lemma-dimension-prime-polynomial-ring}),
    hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim(X)$
    (Algebra, Lemma \ref{algebra-lemma-finite-gl-dim-finite-dim-regular})
    which implies that $\text{Tor}$-groups of modules vanish beyond the dimension
    (More on Algebra, Lemma \ref{more-algebra-lemma-finite-gl-dim-tor-dimension}).
    \end{proof}

    Comments (1)

    Comment #2779 by Ko Aoki on August 19, 2017 a 3:28 pm UTC

    Typo in the proof: "he local rings" should be replaced by "the local rings".

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